9. Locus: Introduction (Part 1)

Welcome to lesson 9. In this lesson, we’ll start with a new topic – Locus.

This is going to be long, so I’ll divide the lesson into two parts, for ease of readability.

Locus is an important part of the coordinate geometry curriculum, and questions involving it are seen throughout all the chapters (which I mentioned in the introductory lesson). Lets begin.

Locus of a Point

Consider a point, moving on your computer screen. If I apply any restriction on the movement of this point (we’ll see in a minute what this restriction can be), then the path traced by this point is known as its locus.

Hmm.. what does that mean? Lets take a very simple example. Suppose that point on the screen is allowed to move such that it remains at a fixed vertical distance from the bottom of your screen (say 5cm). Then what will be its path like? See the following figure.

Fig. 1: The moving point

The point cannot move upwards, as the condition requires that the vertical distance be fixed. Neither it can move downwards, as the distance will decrease. Well? It can of course move sideways, either leftwards or rightwards, parallel to the bottom edge of your screen. Therefore the path traced by the point under this restriction / condition will be a line parallel to the bottom edge of the screen. Or the locus of this point, under the given condition, will be a line (drawn in blue). Have a look.

Fig. 2: The locus – a straight line

Let’s take another example. This time, lets come back to the Cartesian plane. Consider a point moving on the plane, such that it always remains at a fixed distance (say 4) from the origin. That is the point should move in the plane, without ever coming closer to the origin or moving away from it. What do you think?

Fig. 3: What do you think?

Hmm.. sounds familiar to me. The point should move on a circular path !

Fig. 4: The locus – a circle

All points on the circle’s circumference are at the given fixed distance (its radius) from the origin (its center). So, the locus of the point such that it is always at a fixed distance from a given point (in the above case, the origin) is a circle with its radius as the fixed distance and center as the given fixed point.

Hope you haven’t run away. Stay with me. Another example!

What should be the locus of a point, moving in a plane, such that it is at the same distance (i.e. equidistant) from two fixed points?

I don’t care what the distance is (and it is not fixed), my only concern is that if P is the moving point, and A and B are the fixed points, then for all positions of P, the distances should be equal, i.e PA = PB. One obvious point to start with is the midpoint of AB. Let’s take a look at the figure.

Fig. 5: Mystery locus

Now, intuitively I can tell that if the point P moves towards A (or B), then PA (or PB) will decrease. The figure makes it clear. PA becomes smaller than PB if P moves leftwards and larger if it moves rightwards. But the condition says that PA should always be equal to PB. When is that possible? When P moves neither towards A nor towards B, that is along a direction perpendicular to AB. Therefore the locus of P will be the perpendicular bisector of AB.

To summarize, the locus of a point moving in a plane such that it is equidistant from two fixed points, will be the perpendicular bisector of the line segment joining the two points.

Fig. 6: The locus – perpendicular bisector

Note that we can easily prove PA = PB for any point P lying on the perpendicular bisector of AB. Please do it as an exercise, if you’re unable to figure that out.